Erdős problem #233 — wave5n

Checked 2026-07-26. Outcome: PARTIAL, not a solution of the asymptotic problem. The new verifiable output here is an exhaustive finite theorem through \(N=10^7\), a current-literature update missing from the live page summary, and two exact reductions identifying the remaining large-gap tail.

Throughout, claims are labelled as requested:

• (a) elementary-rigorous;
• (b) rigorous modulo the explicitly named theorem (or conditional

theorem);

• (c) plausible/structural-unverified;
• (d) computational-only.

0. Mandatory live-page audit before doing mathematics

Direct datacenter access returns the expected Cloudflare wall. I fetched the rendered page and its discussion thread through the Bright Data residential browser, saved full-page screenshots, opened the discussion URL, and clicked “Show 1 more comments” so that all 16 comments were actually loaded.

Live URL: <https://www.erdosproblems.com/233>

The page displayed OPEN, “0 claimed proofs for this problem,” “Interested in collaborating: None,” and “Currently working on this problem: None.” It also displayed “This problem looks difficult: TerenceTao” and no tractable/formalisation workers. Thus none of the mandatory stop conditions applied. The page says it was last edited 18 January 2026; the discussion has later comments through 31 May 2026.

Verbatim live statement

Let \(d_n=p_{n+1}-p_n\), where \(p_n\) is the \(n\)th prime. Prove that \[ > \sum_{1\leq n\leq N}d_n^2\ll N(\log N)^2. > \]

The live page lists these known results:

• Cramér [Cr36] proved \(O(N(\log N)^4)\) conditional on RH.
• Selberg [Se43], still on RH, proved

\[ \sum_{1\le n\le N}\frac{d_n^2}{n}\ll(\log N)^4. \]

• The PNT gives the matching-order lower bound

\(\sum_{n\le N}d_n^2\gg N(\log N)^2\).

• The desired result would imply

\(d_n\ll n^{1/2}\log n\) for every \(n\); the page says this is known only under RH.

• It points to OEIS A074741 and Guy, problem A8.

These are treated as the page-supplied ground truth, as required.

All 16 live comments read

The site explicitly warns that comments are unverified. The following is a content audit, not an endorsement:

1. Desmond Weisenberg, 19 Aug 2025: located Erdős's page 440 formulation

and Cramér's conditional argument; the page was updated.

1. Terence Tao, 12 Oct 2025: supplied Selberg's weighted RH improvement;

the page was updated.

1. Bhavya Patwa, 18 Jan 2026 07:08: posted an AI-generated proposal using

prime-free intervals and a uniform Hardy–Littlewood assumption.

1. Nat Sothanaphan, 07:26: warned that the AI conversation treated the

problem as a known exercise and was likely hiding a gap.

1. Boris Alexeev, 09:10: pointed to Heath-Brown's constant-\(2\)

conjecture and to Theorem 8 of Funkhouser–Goldston–Ledoan (FGL), whose missing input is control of very large gaps.

1. Bhavya Patwa, 09:21: said the revised AI proof was conditional on a

uniform Hardy–Littlewood \(k\)-tuple conjecture.

1. Bhavya Patwa, 09:29: accepted that the attempted implication had not

dealt with the literature's missing tail.

1. Boris Alexeev, 09:44: observed that conditional implications can still

be useful, with appropriate care.

1. Nat Sothanaphan, 11:49: clarified that ordinary Hardy–Littlewood was

not shown to imply the problem; “HL + large gap tail” is what FGL gives.

1. old-bielefelder, 16:13: apologised for an earlier AI confrontation;

no mathematical result was claimed.

1. Terence Tao, 16:41: stressed the pointwise consequence

\(d_n\ll n^{1/2}\log n\), and that even strong prime-tuple conjectures do not supply it without a separate RH-type input.

1. Will Sawin, 27 May 2026: described a function-field analogue which he

believes reaches only \(O(N(\log N)^3)\), up to the range where every short interval is known to contain a prime.

1. ssssasasaaasrhs, 31 May 2026 08:51: proposed

\(O(N(\log N)^3)\) from a weak Cramér-type gap bound and Pólya–Szegő.

1. Thomas Bloom, 10:28: asked whether the problem is reasonably viewed

as “at least as hard as RH,” while noting that no implication to RH is known.

1. StijnC, 10:34: claimed the weak Cramér assumption makes the desired

bound immediate; the displayed sentence says \(p_n\) is bounded by a logarithmic square, so I do not use this unverified comment.

1. Will Sawin, 12:34: agreed that no implication to RH is expected, but

that a distributional prime problem not implied by RH and with no viable unconditional route can reasonably be called harder.

Importantly, the AI proposal is only a discussion comment: the live database still reports 0 claimed proofs.

1. Primary-source literature check

Original and conditional history

The scan of Erdős's original paper, “The difference of consecutive primes” (1940), page 440, asks in prime-value notation

By the PNT this is the same scale as the live page's index formulation. (b, using the PNT)

I verified that Cramér's paper exists as H. Cramér, Acta Arith. 2 (1936), 23–46. The final page of the scan gives RH-conditional square-gap consequences. FGL Sections 4–5 reproduce the Cramér/Selberg implications in modern notation, including Selberg's \(\mathcal C(x)\ll x\log^3x\) and weighted \(\mathcal S(x)\ll\log^3x\) on RH.

D. R. Heath-Brown, Acta Arith. 41 (1982), 85–99, page 87, explicitly conjectures

(c, Heath-Brown's conjecture) His Corollary 3 gives only \(O(x\log^2x)\) under RH plus a bounded pair-correlation hypothesis, still one logarithm above the target.

The exact conditional obstruction

Funkhouser, Goldston and Ledoan, arXiv:1802.07609, define

Their Theorem 8 says, conditional on their ordinary Hardy–Littlewood hypothesis and their Strong Hardy–Littlewood hypothesis for \(2\le k\le4\), uniformly for tuples in \([1,x^{1/4-\delta}]\), that

This formula was checked against the PDF, not inferred from the comment. FGL explicitly notes that its strong tuple hypothesis cannot rule out the remaining rare very long gaps. (b, conditional on the stated FGL hypotheses)

Best unconditional full second-moment bound found

Maynard's arXiv:1201.1787 records the earlier Peck bound

The arXiv record itself notes that Maynard's argument reproduced Peck's existing result.

The material update absent from the live #233 summary is Julia Stadlmann, “On the mean square gap between primes,” arXiv:2212.10867. Theorem 1 of the 71-page primary PDF proves

Putting \(x=p_N\), using \(p_N\ll N\log N\), and absorbing the logarithm into an arbitrarily small power gives, for every \(\eta>0\),

(b, rigorous modulo Stadlmann's Theorem 1 and the standard explicit upper bound for \(p_N\))

Olli Järviniemi's simultaneous arXiv:2212.10965 improves some first-moment tails for gaps at least \(x^{0.45}\) or \(x^{1/2}\), but its stated theorems do not improve (1) for the complete square sum.

I searched by the exact square-sum formula, the Maynard/Peck exponent, the Stadlmann title and exponent, and recent large-gap papers. I found no later primary source proving the target or improving the full \(x^{1.23+\epsilon}\) bound. This is a reported search miss, not a claim that no unindexed or unpublished improvement exists.

2. Exact elementary reductions

Write

2.1 Integer layer cake

For an integer \(h\ge1\), let

Since \(g^2=\sum_{h=1}^g(2h-1)\) for every positive integer \(g\),

This is an exact finite identity. (a)

2.2 Exact prime-free-interval identity

For \(h\ge1\), define

Inside the gap \((p_n,p_{n+1})\), the valid starting integers are

so there are exactly \((d_n-h)_+\). Hence

Summing over \(h\) and using \(\sum_{h=1}^{g-1}(g-h)=g(g-1)/2\) gives

Thus #233 is equivalently an \(O(N\log^2N)\) bound for the total number, over all lengths, of these prime-free intervals; the telescoping term is only \(O(N\log N)\). (a)

2.3 An iff dyadic tail criterion

Let \(L=\log N\), and for \(j\ge0\) let

The gaps below \(L\) contribute at most \(NL^2\), while each \(j\)-bucket contributes between \(4^jL^2B_j(N)\) and \(4^{j+1}L^2B_j(N)\). Consequently

The constants in the two directions differ by at most \(4\). (a)

This makes the obstruction precise: fixed-\(\lambda\) Poisson laws for \(d_n\sim\lambda\log N\) do not give the uniform summability in (5) as \(j\) grows. A single exceptional gap can dominate a square moment.

3. Exact computation through \(10^7\)

Standalone verifier: erdos233_wave5n_reverify.py

Run from the repository root:

python3 runs/erdos233_wave5n_reverify.py

The clean default run took 61.4 seconds here. It uses only the Python standard library and performs two unrelated prime enumerations:

1. a monolithic odd-only Eratosthenes sieve;
2. an independently written segmented sieve whose base primes come from

trial division.

The passes compare every checkpoint, the complete gap histogram, and an order-sensitive digest of all \(10^7\) gaps. Both gave

gap SHA-256       33d7700aeb812181cd82a6222d27a2876a9ab0fe6afac0ce7f7a28ef83cfaab5
histogram SHA-256 64c2c2dfc368291327bd2e19756d0563e65b2fbc0c08ddc0e29e30cd8cd5526f
frozen_default_output=PASS

Exact checkpoint table

Here \(G_N=\max_{n\le N}d_n\). The ratio column is diagnostic; all integer entries are exact.

The record gap in this range is

Exhaustive finite inequalities

The checker does not use floating point to certify these. It proves rational lower bounds for logarithms from

and, after writing \(n=2^km\),

Finite positive partial sums, rounded downward at scale \(10^{12}\), are rigorous lower bounds. Every comparison is then an integer inequality. The exhaustive result is

and the sharper restricted result is

The certifying pass reports zero failed comparisons; the second pass independently validates the entire gap sequence and all resulting sums. (d)

For orientation only, a floating scan places the largest ratio for \(10\le N\le10^7\) at \(N=11\), about \(2.229287107735\). For \(100\le N\le10^7\), it places the largest at \(N=9,751,264\), about \(2.149452589274\). These locations are not used in (6)–(7).

Independent checks of the reductions

At \(N=10^7\), the program obtains

and

It therefore rechecks (4) as the exact integer equality

It separately reconstructs (3) and (4) from the completed gap histogram.

Some exact tail data from that histogram are:

These are finite data only, not evidence for a uniform asymptotic theorem. (d)

4. Exactly what remains

In prime-value notation the target is

The clean unconditional theorem found is only \(\mathcal C(x)\ll_\epsilon x^{1.23+\epsilon}\). The exponent gap \(0.23\) cannot be repaired by a better treatment of logarithms.

Under the strong tuple hypotheses in FGL, everything except

has already been evaluated. The exact missing lemma would be

for the upper-bound problem, or \(T_\delta(x)=o(x\log x)\) for the constant-\(2\) asymptotic. Ordinary fixed-\(k\) Hardy–Littlewood information does not provide (8), because it has no uniform control over a rare gap whose length grows this far. (b as a deduction from FGL; (8) itself is open)

No finite extension of the computation supplies that uniformity. Runtime is essentially linear in the number of gaps: extrapolating the measured run, \(10^8\) gaps with the same dual audit would cost roughly \(0.2\) core-hours and the monolithic odd sieve would need about \(1.1\) GB; \(10^9\) gaps would be of order \(2\) core-hours and \(12\) GB. Neither finite range addresses (8), so those heavier runs were not made. (c, linear extrapolation from the measured run)

The standard machinery therefore stalls at a sharply named point, rather than at an unspecified “probably open” barrier: it lacks a uniform square-summable upper tail for exceptional prime-free intervals/very large prime gaps.

PARTIAL: Exact dual-sieve verification proves \(C_N\le2.23N(\log N)^2\) for every \(10\le N\le10^7\) (and \(2.15\) for \(100\le N\le10^7\)); asymptotically the unresolved input is the large-gap tail (8), while the best full unconditional bound found is Stadlmann's \(N^{1.23+\epsilon}\).